Lie Ideals and Generalized Derivations. in -Prime Rings - II

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1 International Journal of Algebra, Vol. 6, 2012, no. 29, Lie Ideals and Generalized Derivations in -Prime Rings - II M. S. Khan Department of Mathematics and Statistics Faculty of Science, Sultan Qaboos University P.O. Box36, PCode 123, AL-Khod Muscat, Sultanate of Oman mohammad@squ.edu.om M. A. Khan Department of Mathematics and Computer Sciences Faculty of Natural & Applied Sciences Umaru Musa Yarádua University P. M. B Dutsin-ma Road Katsina, Nigeria moharram_a@yahoo.com Abstract In continuation of our paper [7], we establish the commutativity of a -prime ring R admitting a generalized derivation F satisfying any one of the following properties: (i) d(x) F(y) = 0, (ii) [d(x),f(y)]=0, (iii) d(x) F(y) x y=0, (iv) [d(x),f(y)] [x, y]=0, (v) (d(x) F(y)) [x, y] = 0, and (vi) [d(x), F(y)] x y = 0, x, y in an appropriate subset of R, where R is a 2-torsion free -prime ring with involution, U a non zero

2 1420 M. S. Khan and M. A. Khan -Lie ideal, I a nonzero nonzero derivation d of R commuting with. -ideal of R, and F a generalized derivation associated with a Mathematics Subject Classification: 16W10, 16W25, 16U80 Keywords: Derivations, Lie ideals, rings with involution, σ-prime rings 1. INTRODUCTION Throughout, R will represent an associative ring with center Z(R). Recall that a ring R is prime if arb = 0 implies a = 0 or b = 0. If R has an involution σ, then R is said to be σ-prime if arb = arσ (b) = 0 implies a = 0 or b = 0. Every prime ring equipped with an involution is σ -prime but the converse need not be true in general. As an example, taking S = R R 0 where R 0 is an opposite ring of a prime ring R with (x, y) = (y, x). Then S is not prime if (0, a) S (a, 0) = 0. But if we take (a, b) S (x, y) = 0 and (a, b) S σ ((x, y)) = 0, then arx yrb = 0 and ary xrb = 0, and thus arx = yrb = ary = xrb = 0. This shows that R is σ-prime (see for reference [7]). An ideal I of R is a σ-ideal if I is invariant under σ i.e. σ (I) = I. An additive subgroup U of R is said to be a Lie-ideal if [u, r] U for all u U and r R. A Lie-ideal U which satisfies σ (U) = U is called a σ-lie ideal. If U is a Lie (resp. σ-lie) ideal of R, then U is called a square closed Lie (resp. σ-lie) ideal if u 2 U for all u U. Oukhtite et al. defined a set of symmetric and skew symmetric elements of R as Sa σ (R) = {x R σ(x) = x}. We need the basic commutator identities. For any x, y, z R, [x, yz] = y[x, z] + [x, y]z, [xy, z] = x[y, z] + [x, z]y x ο (yz) = (x o y)z y[x, z] = y(x o z) + [x, y]z (xy) o z = x(y o z) [x, z]y = (x o z)y + x[y, z]. An additive map F: R R is called a generalized derivation associated with d if there exists a derivation d: R R (an additive map d: R R is called a derivation if d (xy) = d(x)y + xd(y) holds for all x, y R) such that F (xy) = F(x) y + xd(y) for all x, y R( Bresar [4]). Clearly, the notion of generalized derivation covers the notions of a derivation and a left multiplier (i.e. F (xy) = F(x) y, for all x, y R). Particularly, one can observe that: For a fixed a R, the map d a : R R defined by d a (x) = [a, x] for all x R is a derivation which is said to be an inner

3 Lie ideals and generalized derivations 1421 derivation. An additive map g a,b : R R is called a generalized inner derivation if g a,b (x) = ax + xb for some fixed a, b R. It is easy to see that if g a,b (x) is a generalized inner derivation, then g a,b (x) (xy) = g a,b (x)y + xd b (y) for all x, y R, where d b is an inner derivation. Several authors [1, 2, 3, 16, 18] have established a vast theory concerning derivations and generalized derivations of prime rings and are still adding. In 2005, Oukhtite et al. gave rise to an extension of prime rings in the form of -prime rings and proved numerous results which hold true for prime rings (see for references [7-15]). In their papers [19, 20], Huang contributed to the newly emerged theory by extending the results of ([2, 3]). In this context, Khan et al.([5, 6]) extended results concerning derivations and generalized derivations of -prime rings to some more general settings. In [1], Ashraf et al. studied the commutativity of a prime ring R admitting a generalized derivation F with associated derivation d satisfying any one of the conditions: (i) d(x) F(y) = 0, (ii) [d(x), F(y)] = 0, (iii) d(x) F(y) x y = 0, (iv) (d(x) F(y)) [x, y] = 0, (v) [d(x), F(y)] [x, y] = 0, and (vi) [d(x), F(y)] x y = 0 for all x, y I, where I is non zero ideal of prime ring R. In 2007, Huang [21] obtained similar results by considering Lie ideals instead. A continuous approach in the direction of -prime rings is still on. However, results concerning inner derivations and generalized inner derivations in -prime rings are still naive. In the present paper, we develop some more results in this regards, Further, we extend these results for - prime rings. 2. PRELIMINARY RESULTS We begin with Lemma 2.1([15, Lemma 4]) If U Z(R) is a -Lie ideal of a 2-torsion free σ-prime ring R and a, b R such that aub = (a)ub = 0 or aub = au(b), then a = 0 or b = 0. Lemma 2.2([13, Theorem 1.1])Let R be a 2-torsion free σ-prime ring, U a nonzero Lie ideal of R and d a nonzero derivation of R which commutes with σ. If d 2 (U) = 0,then U Z(R). Lemma 2.3 ([13, Lemma 2.3]) Let 0 U be a -Lie ideal of a 2-torsion free -prime ring R. If [U, U] = 0, then U Z(R). Lemma 2.4 ([11, Lemma 2.2]) Let R be a 2-torsion free -prime ring and U a nonzero -Lie ideal of R. If d is a derivation of R which commutes with and satisfies d (U) = 0, then either d = 0 or U Z(R).

4 1422 M. S. Khan and M. A. Khan Now, we establish the following lemma Lemma 2.5 Let R be a 2-torsion free -prime ring and U a nonzero -Lie ideal of R. If d is a derivation of R which commutes with such that d(x) o y = 0 for all x, y U, then either d = 0 or U Z(R). Proof: Let U Z(R). We have d(x) o y = 0 for all x, y U. (1) Replacing y by yz in (1), we get y [d(x), z] = 0 for all x, y, z U. (2) Putting y by d 2 (x) y in (2), we obtain d 2 (x) y [d(x), z] = 0 for all x, y, z U. or d 2 (x) U [d(x), z] = 0 for all x, z U. (3) The relation (3) is same as obtained in proof of the [20, Theorem 3.2]. Hence, we get the result d = MAIN RESULTS Theorem 3.1: Let R be a 2-torsion free -prime ring and U a nonzero square closed - Lie ideal of R. Suppose there exists a generalized derivation F associated with a nonzero derivation d, commuting with, such that d(x) F(y) = 0, for all x, y U, then U Z(R). Proof: Suppose U Z(R). We have d(x) F(y) = 0 for all x, y U. (4) Replacing y by yz in (4) and using (4), we obtain (d(x) y) d(z) y[d(x), d(z)] F(y)[d(x), z] = 0 for all x, y, z U. (5) Now replacing z by zd(x) in (5), yields (d(x) y)) d(z d(x)) y[d(x), z d(x)] F(y)[d(x), z d(x)] = 0, or (d(x) y)){d(z) d(x) + z d 2 (x)} y[d(x), z ]d(x) F(y)[d(x), z] d(x) = 0 for all x, y, z Using (5) in the above obtained relation (6), implies (d(x) y)) z d 2 (x) = 0 for all x, y, z U or (d(x) y)) U d 2 (x) = 0 for all x, y U. (7) U.(6)

5 Lie ideals and generalized derivations 1423 Let x U (R). Since d commutes with, application of Lemma 2.1 in (7), yields d(x) y = 0 or d 2 (x) = 0 for all y U. (8) Let x U. Since x (x) U (R), from (8) it follows that d(x (x)) y = 0 or d 2 (x (x)) = 0 for all y U. Case 1: Suppose d(x (x)) y = 0; if d(x + (x)) y = 0, then (d(x) d( (x)) + d(x) + d( (x))) o y = 0 or 2d(x) o y = 0. Since char R 2, we get d(x) o y = 0 for all x, y U. Case 2: Now suppose that d 2 (x (x)) = 0. Since d commutes with d(d(x) d( (x))) =0 or d 2 (x) d( (d(x))) =0 or Thus, d 2 (x) d 2 (x) = (d 2 (x)). (R) and again by virtue of Lemma 2.1 and (7), we get, we have d(x) y = 0 or d 2 (x) = 0. In conclusion, for all x U we have either d(x) U = 0 or d 2 (x) = 0. Accordingly, U is a union of two additive subgroups G and H, where G = {x U d(x) o y = 0, for all y U} and H = {x U d 2 (x) = 0}. But a group cannot be a union of two of its subgroups and thus U = G or U = H. If U = G, then by virtue of Lemma 2.5 we get either d = 0 or U Z(R). If U = H, then by application Lemma 2.2 we get either d = 0 or U Z(R). Hence, both the cases yields either d = 0 or U Z(R). But, since d is non zero, then U must be contained in Z(R). Theorem 3.2: Let R be a 2-torsion free -prime ring and U a nonzero square closed - Lie ideal of R. Suppose there exists a generalized derivation F associated with a nonzero derivation d, commuting with, such that [d(x), F(y)] = 0, for all x, y U, then U Z(R). Proof: Suppose U Z(R). We have [d(x), F(y)] = 0 for all x, y U. (9) Replacing y by yz in (9) and using (9), we obtain F(y)[d(x), z] + [d(x), y] d(z) + y[d(x), d(z)] = 0 (10) Now replacing z by zd(x) in (10), yields [d(x), y] z d 2 (x) + y [d(x), zd 2 (x)] = 0 (11)

6 1424 M. S. Khan and M. A. Khan Substituting y by wy in (11) and applying (11), we have [d(x), w] yz d 2 (x) = 0 or [d(x), w] U U d 2 (x) = 0 for all x, w U. (12) Let x U (R). Since d commutes with, application of Lemma 2.1 in (12), yields [d(x), w] = 0 or U d 2 (x) = 0 for all w U. Let x U. Since x (x) U (R), from (12) in combination with Lemma 2.1, it follows [d(x (x)), w] = 0 or d 2 (x (x)) = 0 for all w U. Case 1: Suppose [d(x (x)), w] = 0 or [d(x), w] = [d( (x)), w] = ([d(x), w]) Hence by (12) we get [d(x), w] = 0 or d 2 (x) = 0 for all x U. Case 2: Suppose d 2 (x (x)) = 0. Using the same arguments as used in proof of Case 2 in Theorem 3.1, we have [d(x), w] = 0 or d 2 (x) = 0 for all x U. (13) Using the same techniques as used in the proof of [20, Theorem 3.2] from (13), we conclude that d = 0. Since d is given to be non zero, a contradiction is obtained. Thus, U must be contained in Z(R). Remark 1: The technique used in proof of Theorem 3.2 from equation (12) onwards till end would be needed further in the paper. We name this portion as result A Theorem 3.3: Let R be a 2-torsion free -prime ring and U a nonzero square closed - Lie ideal of R. Suppose R admits a generalized derivation F associated with a derivation d, commuting with, such that d(x)o F(y) = x o y, for all x, y U. If F = 0 or d 0 then U Z(R). Proof: If F = 0, then x o y = 0 for all x, y U. (14) For any z U replacing y by yz in (14), we get x o yz = 0 or (x o y)z y[x, z] = 0 or y [x, z] = 0 or U[x, z] =0

7 Lie ideals and generalized derivations 1425 or 1 U[x, z] =0 or (1) U [x, z] =0, since U is -Prime Lie ideal of R, we have [x, z] = 0 for all x, z U. (15) In view of Lemma 2.3, (15) implies that U Z(R). Now assume that d 0. We have d(x) o F(y) = x o y for all x, y U. (16) For any z U, replacing y by yz in (16), we get (d(x) o y) d(z) y [d(x), d(z)] F(y)[d(x), z] + y[x, z] = 0 (17) Replacing z by zd(x) in (17) and applying (17), we obtain (d(x) o y) z d 2 (x) y[d(x), zd 2 (x)] + yz [x, d(x)] = 0 (18) Substitute y by zy in (18) and using (18), we have [d(x), z] U d 2 (x) = 0 for all x, z U. (19) Consequently, by application of result A in (19), we get U Z(R). Now using similar techniques, one can also prove the following theorem. Theorem 3.4: Let R be a 2-torsion free -prime ring and U a nonzero square closed - Lie ideal of R. Suppose R admits a generalized derivation F associated with a derivation d, commuting with, such that d(x)o F(y)+ x o y = 0, for all x, y U. If F = 0 or d 0 then U Z(R). Theorem 3.5: Let R be a 2-torsion free -prime ring and U a nonzero square closed - Lie ideal of R. Suppose R admits a generalized derivation F associated with a derivation d, commuting with, such that [d(x), F(y)] ± [ x, y] = 0, for all x, y U, then U Z(R). Proof: (i) Suppose U Z(R). We have [d(x), F(y)] - [ x, y] = 0 for all x, y U. (20) Replacing y by yz in (20), we get [d(x), F(y)]z + F(y)[d(x), z] + [d(x), y]d(z) + y[d(x), d(z)] = y[x, z] + [x, y]z or F(y)[d(x), z] + [d(x), y]d(z) + y[d(x), d(z)] = y[x, z] for all x, y, z U.(using 20) (21) Substituting z by zd(x) in (21) and applying (21), we obtain [d(x), y] z d 2 (x) + yz [d(x), d 2 (x)] + y [d(x), z] d 2 (x) yz [x, d(x)] = 0 (22) Replacing y by wy in (22) and using (22), we have [d(x), w] yz d 2 (x) = 0 for all x, y, z, w U or [d(x), w] U d 2 (x) = 0 for all x, w U. (23) Hence by application of result A we conclude the result. (ii) Using the similar technique with necessary variations, the result follows.

8 1426 M. S. Khan and M. A. Khan Theorem 3.6 Let R be a 2-torsion free -prime ring and U a nonzero square closed - Lie ideal of R. Suppose R admits a generalized derivation F associated with a derivation d, commuting with, such that d(x) o F(y) ± [ x, y] = 0, for all x, y U, then U Z(R). Proof: (i) Suppose U Z(R). We have d(x) o F(y) - [ x, y] = 0, for all x, y U (24) Replacing y by yz in (24) and using (24),we get - F(y) [d(x), z] + (d(x) o y) d(z) y[d(x), d(z)] = y[x, z] (25) Substituting z by zd(x) in (25) and applying (25), we obtain (d(x) o y) z d 2 (x) yz [d(x), d 2 (x)] y [d(x), z] d 2 (x) = yz [x, d(x)] (26) For any w U, replacing y by wy in (26) and using (26), we have [d(x), w] y z d 2 (x) = 0 for all x, y, z U or [d(x), w] U d 2 (x) = 0 for all x, w U. (27) Reasoning as used in proof of result A, we conclude the result. (ii) Using the similar technique as used in (i) with necessary variations, the result follows. Theorem 3.7 Let R be a 2-torsion free -prime ring and U a nonzero square closed - Lie ideal of R. Suppose R admits a generalized derivation F associated with a derivation d, commuting with, such that[d(x), F(y] ± x o y = 0, for all x, y U, then U Z(R). Proof: Suppose U Z(R). We have [d(x), F(y)] = x o y, for all x, y U (28) Replacing y by yz in (28) and using (28), we get F(y) [d(x), z] + [d(x), y] d(z) + y[d(x), d(z)] = - y[x, z] (29) Substituting z by zd(x) in (29) and applying (29), we obtain yz [d(x), d 2 (x)] + y [d(x), z] d 2 (x) + [d(x), y] z d 2 (x) = - yz [x, d(x)] (30) For any w U, replacing y by wy in (30) and using (30), we have [d(x), w] y z d 2 (x) = 0 for all x, y, z U or [d(x), w] U d 2 (x) = 0 for all x, w U. (31) Applying result A, we get the desired result. (ii) Using the similar technique as used in (i) with necessary variations, the result follows.

9 Lie ideals and generalized derivations 1427 In view of these results we get the following corollary: Corollary 3.7: Let R be a 2-torsion free -prime ring and I a nonzero -ideal of R. Suppose that R admits a generalized derivation F associated with a nonzero derivation d, commuting with, such that any of the following holds: (i) d(x) F(y) = 0, (ii) [d(x),f(y)] = 0, (iii) d(x)o F(y) = x o y, (iv) d(x)o F(y) ± x o y = 0, (v) [d(x), F(y)] ± [ x, y] = 0 for all x, y I. Then R is commutative. REFERENCES [1] M. Ashraf, A. Ali, and R. Rani, On generalized derivations of prime rings, Southeast Asian Bulletin of Mathematics, vol. 29, no. 4, pp , [2] M.Ashraf, A.Ali, S.Ali, Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull.Math., 31(2007), [3] M. Ashraf & N. Rehman, On commutativity of rings with derivations, Result. Math., 42 (2002), 3 8. [4] M. Bresar, On the distance of the compositions of two derivations to the generalized derivations, Glasgow Math. J. 33 (1991), no. 1, [5] Khan. M.R, Arora. D and Khan. M.A, 2010, Lie derivations and generalized derivations in σ-prime rings (Communicated in Algebra Colloquium). [6] Khan. M.R, Arora. D and Khan. M.A, 2010, Remarks on Derivations of σ -prime rings, International Journal of Algebra,4, 2010, no. 16,

10 1428 M. S. Khan and M. A. Khan [7] Khan. M.R, Arora. D and Khan. M.A, 2010, Notes on derivations and Lie ideals in σ -prime rings, Advances in Algebra, 3, no. 1 (2010), pp [8] Oukhtite. L & Salhi. S, 2006, On generalized derivations of -prime rings, African Diaspora J. Math., 5, no. 1, pp [9] L. Oukhtite and S. Salhi, On commutativity of σ-prime rings, GlasnikMathematicki, 41(2006), [10] L. Oukhtite and S. Salhi, On derivations in σ-prime rings, Int. J. Algebra, 1(2007), [11] L. Oukhtite and S. Salhi, Derivations and commutativity of σ-prime rings, Int. J. Contemp, 1(2006), [12] L. Oukhtite and S. Salhi, σ-lie ideals with derivations as homomorphisms and antihomomorphisms, Int. J. Algebra, 1(2007), [13] L. Oukhtite, S. Salhi and L. Taoufiq, On generalized derivations and commutativity in σ- prime rings, Int. J. Algebra, 1(2007), [14] L. Oukhtite and S. Salhi, Lie ideals and derivations of σ- prime rings, Int. J. Algebra, 1(2007), [15] L. Oukhtite and S. Salhi, On generalized derivations of σ-prime rings, Afr. Diaspora J. Math., 5(2006), [16] L. Oukhtite and S. Salhi, Centralizing automorphisms and Jordan Left derivations on σ- prime rings, Advances in Algebra, 1(2008), no. 1, [17] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), [18]M. A. Quadri, M. Shadab Khan and N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math. 34 (2003), no. 9,

11 Lie ideals and generalized derivations 1429 [19] N. Rehman, R.M. Omary and C. Haetinger, On Lie structure of prime rings with generalized (,) derivations, Bol. Soc. Paranaense Math., 27 (2009), no. 2, [20] H. Shuliang, Generalized derivations of σ-prime rings, Int. J. of Algebra, 2 (2008), no. 18, [21] H. Shuliang, Lie ideals and derivations of σ-prime rings,j. Korean Soc. Math. Educ. Ser. B Pure Appl. Math.,17(2010),no.1, [22] H. Shuliang, Generalized Derivations of Prime Rings, International Journal of Mathematics and Mathematical Sciences, Volume Received: September, 2012